If electron velocity is `(2 hat(i) + 3 hat(j))` and it is subjected to a magnetic field 4 K then

Options

A. speed will change
B. path will be curved
C. Both A and B
D. None of the above

Answer

To solve the problem, we need to analyze the motion of an electron with a given velocity in a magnetic field. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
The velocity of the electron is given as:
\[ \vec{v} = 2\hat{i} + 3\hat{j} \]
This indicates that the electron is moving in the x-y plane.

The magnetic field is given as:
\[ \vec{B} = 4\hat{k} \]
This means the magnetic field is directed along the positive z-axis.

### Step 2: Calculate the Magnitude of the Velocity
To find the magnitude of the velocity, we use the formula:
\[ |\vec{v}| = \sqrt{(v_x)^2 + (v_y)^2} \]
Substituting the values:
\[ |\vec{v}| = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Determine the Direction of the Velocity
To find the angle θ that the velocity vector makes with the x-axis, we can use the tangent function:
\[ \tan(\theta) = \frac{v_y}{v_x} = \frac{3}{2} \]
Thus,
\[ \theta = \tan^{-1}\left(\frac{3}{2}\right) \]
Calculating this gives approximately:
\[ \theta \approx 56.31^\circ \]

### Step 4: Analyze the Motion in the Magnetic Field
When a charged particle (like an electron) moves in a magnetic field, it experiences a magnetic force given by:
\[ \vec{F} = q(\vec{v} \times \vec{B}) \]
where \( q \) is the charge of the electron. The direction of the force is perpendicular to both the velocity and the magnetic field.

### Step 5: Determine the Effects of the Magnetic Field
Since the magnetic force is always perpendicular to the velocity of the electron, the speed of the electron will remain constant, but its direction will change. This results in a curved path (specifically, circular motion if the magnetic field is uniform).

### Step 6: Conclusion
From the analysis:
- The speed of the electron does not change.
- The path of the electron will be curved due to the magnetic force acting on it.

Thus, the correct options based on the problem statement are:
- The speed will not change.
- The path will be curved.

### Final Answer
The correct option is: **Both A and B** (the speed remains constant while the path is curved).

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